a What is the impulse response h t of the ideal band stop filter b Describe how

# A what is the impulse response h t of the ideal band

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(a) What is the impulse response h ( t ) of the ideal band-stop filter? (b) Describe how to implement the ideal band-stop filter using only lowpass and highpass filters. 2.6.12 The ideal continuous-time band-pass filter has the frequency response H f ( ω ) = 0 | ω | ≤ ω 1 1 ω 1 < | ω | < ω 2 0 | ω | ≥ ω 2 . Describe how to implement the ideal band-pass using two ideal low-pass filters with different cut-off frequencies. (Should a parallel or cascade structure be used?) Using that, find the impulse response h ( t ) of the ideal band-pass filter. 2.6.13 The ideal continuous-time high-pass filter has the frequency response H f ( ω ) = 0 , | ω | ≤ ω c 1 , | ω | > ω c . Find the impulse response h ( t ). 2.6.14 The frequency response of a continuous-time LTI is given by H f ( ω ) = 0 | ω | < 2 π 1 | ω | ≥ 2 π (This is an ideal high-pass filter.) Use the Fourier transform to find the output y ( t ) when the input x ( t ) is given by (a) x ( t ) = sinc( t ) = sin( π t ) π t (b) x ( t ) = sinc(3 t ) = sin(3 π t ) 3 π t 2.6.15 What is the Fourier transform of x ( t )? x ( t ) = cos(0 . 3 π t ) · sin(0 . 1 π t ) X f ( ω ) = F{ x ( t ) } = ? 57
2.7 Matching 2.7.1 Match the impulse response h ( t ) of a continuous-time LTI system with the correct plot of its frequency response | H f ( ω ) | . Explain how you obtain your answer. -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -1 -0.5 0 0.5 1 t IMPULSE RESPONSE -8 -6 -4 -2 0 2 4 6 8 0 0.2 0.4 0.6 0.8 1 ω FREQUENCY RESPONSE A -8 -6 -4 -2 0 2 4 6 8 0 0.2 0.4 0.6 0.8 1 ω FREQUENCY RESPONSE B -8 -6 -4 -2 0 2 4 6 8 0 0.2 0.4 0.6 0.8 1 ω FREQUENCY RESPONSE C -8 -6 -4 -2 0 2 4 6 8 0 0.2 0.4 0.6 0.8 1 ω FREQUENCY RESPONSE D 58
2.7.2 For the impulse response h ( t ) illustrated in the previous problem, identify the correct diagram of the poles of H ( s ). Explain how you obtain your answer. -3 -2 -1 0 1 2 3 -10 -5 0 5 10 #1 Real part Imag part -3 -2 -1 0 1 2 3 -10 -5 0 5 10 #2 Real part Imag part -3 -2 -1 0 1 2 3 -10 -5 0 5 10 #3 Real part Imag part -3 -2 -1 0 1 2 3 -10 -5 0 5 10 #4 Real part Imag part 59
2.7.3 The following diagrams indicate the pole locations of six continuous-time LTI systems. Match each with the corresponding impulse response with out actually computing the Laplace transform. POLE-ZERO DIAGRAM IMPULSE RESPONSE 1 2 3 4 5 6 -1 0 1 -6 -4 -2 0 2 4 6 #1 -1 0 1 -6 -4 -2 0 2 4 6 #2 -1 0 1 -6 -4 -2 0 2 4 6 #3 -1 0 1 -6 -4 -2 0 2 4 6 #4 -1 0 1 -6 -4 -2 0 2 4 6 #5 -1 0 1 -6 -4 -2 0 2 4 6 #6 0 2 4 -1 0 1 2 A 0 2 4 -1 0 1 2 B 0 2 4 -1 0 1 2 C 0 2 4 -1 0 1 2 D 0 2 4 -1 0 1 2 E 0 2 4 -1 0 1 2 F 60
2.7.4 The figure shows the impulse responses and frequency responses of four continuous-time LTI systems. But they are out of order. Match the impulse response to its frequency response magnitude, and explain your answer. IMPULSE RESPONSE FREQUENCY RESPONSE 1 2 3 4 -1 0 1 2 3 4 5 6 -1 -0.5 0 0.5 1 IMPULSE RESPONSE 1 -10 -5 0 5 10 0 0.2 0.4 0.6 FREQUENCY RESPONSE 1 -1 0 1 2 3 4 5 6 -1 -0.5 0 0.5 1 IMPULSE RESPONSE 2 -10 -5 0 5 10 0 0.2 0.4 0.6 FREQUENCY RESPONSE 3 -1 0 1 2 3 4 5 6 -1 -0.5 0 0.5 1 IMPULSE RESPONSE 3 -10 -5 0 5 10 0 0.2 0.4 0.6 FREQUENCY RESPONSE 2 -1 0 1 2 3 4 5 6 -1 -0.5 0 0.5 1 IMPULSE RESPONSE 4 TIME (SEC) -10 -5 0 5 10 0 0.2 0.4 0.6 FREQUENCY RESPONSE 4 FREQUENCY (Hz) 61
2.7.5 The figure shows the impulse responses and frequency responses of four continuous-time LTI systems.

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